Takahashi (1989) hypothesized that the basic architecture of a chromosome is
tree-like, consisting of a concatenation of 'mini-chromosomes'. A fractal
dimension of D = 2.34 was determined from an analysis of first and second order
branching patterns in a human metaphase chromosome. Xu et al. (1994)
hypothesized that the twistings of DNA binding proteins have fractal properties.

Lewis and Rees (1985) determined the fractal dimension of protein surfaces (2
<= D <= 3) using microprobes. A mean surface dimension of D = 2.4 was determined
using microprobe radii ranging from 1-3.5 angstroms. More highly irregular
surfaces (D > 2.4) were found to be sites of inter-protein interaction. Wagner
et al. (1985) estimated the fractal dimension of heme and iron-sulfur proteins
using crystallographic coordinates of the carbon backbone. They found that the
structural fractal dimension correlated positively with the temperature
dependence of protein relaxation rates.

Smith et al. (1989) used fractal dimension as a measure of contour complexity
in two-dimensional images of neural cells. They recommend D as a quantitative
morphological measure of cellular complexity.

Self-similarity has recently been found in DNA sequences (summarized in
Stanley 1992; see also papers in Nonnenmacher et al. 1994). Glazier et al.
(1995) used the multifractal spectrum approach to reconstruct the evolutionary
history of organisms from m-DNA sequences. The multifractal spectra for
invertebrates and vertebrates were quite different, allowing for the recognition
of broad groups of organisms. They concluded that DNA sequences display fractal
properties, and that these can be used to resolve evolutionary relationships in
animals. Xiao et al. (1995) found that nucleotide sequences in animals, plants
and humans display fractal properties. They also showed that exon and intron
sequences differ in their fractal properties.

The kinetics of protein ion channels in the phospholipid bilayer were
examined by Liebovitch et al. (1987). The timing of openings and closings of ion
channels had fractal properties, implying that processes operating at different
time scales are related, not independent (Liebovitch and Koniarek 1992).
López-Quintela and Casado (1989) developed a fractal model of enzyme kinetics,
based on the observation that kinetics is a function of substrate concentration.
They found that some enzyme systems displayed classical Michaelis-Menten
kinetics (D = 1), while others showed fractal kinetics (D < 1).

Fractal dichotomous branching is seen in the lung, small intestine, blood
vessels of the heart, and some neurons (West and Goldberger 1987; Goldberger et
al. 1990; Glenny et al. 1991; Deering and West 1992). Fractal branching greatly
amplifies the surface area of tissue, be it for absorption (e.g. lung,
intestine, leaf mesophyll), distribution and collection (blood vessels, bile
ducts, bronchial tubes, vascular tissue in leaves) or information processing
(nerves). Fractal structures are thought to be robust and resistant to injury by
virtue of their redundancy and irregularity. Nelson et al. (1990) examined
power-law relationships between branch order and length in human, dog, rat and
hamster lung tissue. Differences between the human lung and those of other
species were hypothesized to be related to postural orientation. Long (1994)
relates Leonardo da Vinci's ratio of branch diameters in trees (= 0.707) to
observed dichotomous fractal bifurcations.

Tyler and Wheatcraft (1990) offer a useful overview of the application of
fractal scaling to soil physics. Tyler and Wheatcraft (1989) used particle-size
distributions to determine the fractal dimension of various soils, and to relate
D to such soil properties as percolation and surface water retention. Perfect
and Kay (1991) used a similar method to examine soil fragmentation, while
Bartoli et al. (1991) used various methods to estimate the mass, pore and
surface fractal dimensions of silty and sandy soils. Eghball et al. (1993) used
Rosin's Law to demonstrate that different tillage methods and crop sequences
affected soil fragmentation (fractal dimension). Perfect et al. (1993) modelled
the relationship between soil aggregate size and tensile strength using a
multifractal approach. Frontier (1987: 340) suggests that it would be
interesting to examine the relationship between soil microflora-fauna diversity
and soil fractal geometry.

Vlcek and Cheung (1986) measured the fractal dimension of leaf edges in a
number of species. Although D was found to be highly variable in some species
(e.g. oaks), they felt that D might have potential as a taxonomic character. The
fractal dimension of root systems was examined by Tatsumi et al. (1989) using
the box-counting method. They found fractal dimensions in the range 1.46 - 1.6
for mature crop plants. Fitter and Strickland (1992) demonstrated that the
fractal dimension of root systems increases over time (to a maximum D of
approximately 1.35), and varies between species. Corbit and Garbary (1994) found
no differences in the fractal dimension of three algal species, though D
increased with both developmental stage and frond structural complexity.

Zeide and Gresham (1991) estimated the fractal dimension of the crown surface
of loblolly pine trees in North Carolina, and found evidence that D varies with
site quality and thinning intensity. Osawa (1995) determined that trees with
higher crown fractal dimensions have less negative self-thinning exponents. It
was hypothesized that species-specific changes in foliage packing over time
account for this relationship. Chen et al. (1994) developed a fractal-based
canopy structure model to calculate light interception in poplar stands.

The fractal geometry of fungal foraging is described by Ritz and Crawford
(1990). Fractal dimension varies between fungal species, and tends to be greater
when nutrient availability is higher (Bolton and Boddy 1993).

Nonlinear dynamics is the study of systems that respond disproportionately to
stimuli. A simple deterministic nonlinear system may behave erratically (though
not randomly), a state which has been termed chaos. Chaotic systems are
characterized by complex dynamics, determinism, and sensitivity to initial
conditions, making long-term forecasting impossible. Chaos, which is closely
related to fractal geometry, refers to a kind of constrained randomness (Stone
and Ezrati 1996). Wherever a chaotic process has shaped an environment, a
fractal structure is left behind.

Goldberger et al. (1990) state that physiology may prove to be one of the
richest laboratories for the study of fractals and chaos as well as other types
of nonlinear dynamics. A good example is the study of heart rate time series
(Goldberger 1992). Conventional wisdom states that the heart displays 'normal'
periodic rhythms that become more erratic in response to stress or age. However,
recent evidence suggests just the opposite: physiological processes behave more
erratically (chaotically) when they are healthy and young. Normal variation in
heart rate is 'ragged' and irregular, suggesting that mechanisms controlling
heart rate are intrinsically chaotic. Such a mechanism might offer greater
flexibility in coping with emergencies and changing environments. Lipsitz and
Goldberger (1992) found a loss of complexity in heart rate variation with age.
Based on this result, they defined aging as a progressive loss of complexity in
the dynamics of all physiological systems. Sugihara (1994), using a different
analytical approach, found that prediction-decay and nonlinearity models are
good predictors of human health. Healthy patients have a steeper heart rate
decay curve, and have greater nonlinearity in their heart rhythms. Teich and
Lowen (1994) found that human auditory neuron transmissions are best modelled as
fractal point processes, and that such transmissions display long-term
persistence (H > 0.5). Hahn et al. (1992) examined thermoregulation responses to
heat stress in cattle. Fractal dimensions of thermoregulation profiles were
found to decrease with increasing stress. They also found that the interval
between temperature reading was critical to the detection of changes in
thermoregulatory profiles. Similar results were obtained by Escós et al. (1995)
in a study of stress in wild goats (spanish ibex). These authors also found that
plants under stress show greater variability in allometric relationships, and
reduced branch structure complexity.

Basic ideas of chaotic dynamics in population biology are summarized by
Schaffer and Kot (1986). The question of whether natural population cycles are
deterministic or purely stochastic was examined by Sugihara et al. (1990; also,
Sugihara and May 1990). They state that populations are embedded in a dynamic
web of other species and environmental forces, implying that irregularities in
population cycles (which have traditionally been 'smoothed' prior to modelling)
may provide important information regarding their dynamics. Sugihara et al.
(1990) found that for pure additive noise, the correlation of adjacent values
was independent of the prediction interval, but for chaotic trends correlations
decline as the prediction interval increased. They found that measles epidemics
display chaotic properties, but that chickenpox epidemic patterns are best
modelled as noise superimposed on a strong annual cycle. Ellner and Turchin
(1995) have argued that it is potentially misleading to make a strict
distinction between chaotic and stochastic dynamics. Using an approach of
non-linear time-series modelling and estimation of Lyapunov exponents (see
Godfray and Grenfell 1993), they demonstrated that ecological populations vary
from noise-dominated, stable dynamics to weakly chaotic ones. However, Sugihara
(1994) claims that their approach is fundamentally flawed, and offers an
alternative method based on locally-weighted maps. Hastings et al. (1993)
summarize the various methods available for detecting deterministic chaos in
biological time series.

Sugihara and May (1990) examined persistence (probability of extinction) in
time series of population sizes. The higher the value of H (lower fractal
dimension), the smoother and more persistent the population trend. Higher
persistence (H) makes a species more prone to extinction, since population
values increase (or decrease) faster over time than in populations having low H.
Hastings and Sugihara (1993: 138-160) expand on these ideas, and present
examples based on bird and butterfly population time series.

Stone and Ezrati (1996) discuss potential applications of nonlinear dynamics
and chaos theory to the study of ecological variability. They argue that chaos
theory may be particularly useful in modelling vegetation change, where
non-equilibrium dynamics (e.g. disturbance, natural mosaic cycling, and habitat
fragmentation) often prevail.

Morse et al. (1985) argued that since habitat has a fractal structure, there
will be more 'useable' space for smaller animals than for larger ones. Working
with invertebrates, they found that predictions of the number of individuals (by
size class) based on body mass and metabolic rate alone consistently
underestimated observed field values for the smaller size classes. Predictions
were considerably improved when the fractal dimension of the habitat was
incorporated into the model: smaller organisms 'perceive' more space and are
therefore comparatively more abundant. Shorrocks et al. (1991) confirmed this
general result, as did Gunnarsson (1992) and Jeffries (1993) using artificial
substrates of differing fractal dimension.

Fractional Brownian motion models (Frontier 1987: 351-353) have been used to
characterize the movement of organisms. Dicke and Burrough (1988) used fractal
analysis to examine spider mite movements in the presence and absence of a
dispersing pheromone. Wiens and Milne (1989) took a different approach,
examining beetle movements in natural fractal landscapes. They found that
observed beetle movements deviated from the modelled (fractional Brownian) ones.
A follow-up study by Johnson et al. (1992a) found that beetle movements reflect
a combination of ordinary (random) and anomalous diffusions. The latter may
simply reflect intrinsic departures from randomness, or result from barrier
avoidance and utilization of corridors in natural landscapes. Johnson et al.
(1992b) discuss the interaction between animal movement characteristics and the
patch-boundary features in a 'microlandscape'. They argue that such interactions
have important spatial consequences on gene flow, population dynamics and other
ecological processes in the community (see also Wiens et al. 1995). In a
comparison of path tortuosity in three species or grasshopper, With (1994a)
found that the path fractal dimension of the largest species was smaller than
those of the two smaller ones. She suggested that this reflects the fact that
smaller species interact with the habitat at a finer scale of resolution than do
larger species. In a second study, With (1994b) found differences in the ways
that gomphocerine grasshopper nymphs and adults interacted with the
microlandscape.

Frontier (1987: 337-343) discusses the ecological significance of contact
zones (ecotonal boundaries) between ecosystems, and outlines how fractal theory
can be used to examine boundary phenomena. For example, consider the contact
surfaces created by turbulence in aquatic ecosystems (the geometry of which is
fractal, Mandelbrot 1982; Milne 1988: 72). Turbulent regions (e.g. interfaces
between warm and cold water) have high phytoplankton productivity due to
increased contact with resources (nutrients and light), which in turn 'feeds'
higher trophic levels. This cascade effect implies that spatial patterns at fine
spatial scales determine patterns at broader scales. Pennycuick and Kline (1986)
estimated D to determine bald eagle territory sizes along rocky coastlines in
Alaska. Forest-grassland ecotones could also be examined in this way to
determine habitat available to foraging animals, or to plant species restricted
to ecotonal environments. Ecotone concepts can also be applied to the design of
public spaces (Arlinghaus and Nystuen 1990).

Burrough (1981) used the semivariogram method to estimate D for various
environmental transects (e.g. soil factors, vegetation cover, iron ore content
in rocks, rainfall levels, crop yields). He found high fractal dimensions in all
cases, from D = 1.4 (iron ore content at 3 m intervals) to D = 2.0 (soil pH at
10 m intervals). Very high fractal dimensions indicate spatial independence of
successive values. While some of the series displayed self-similarity over many
scales (i.e. a linear log-log plot slope), other trends suggested variation in D
with changing scale. Palmer (1988) used the same method to examine spatial
dependence of vegetation along transects. Values were generally high but not
scale-invariant. Based on a fractal analysis, Phillips (1985) concluded that
erosion processes along a portion of the Delaware coast could not be easily
predicted.

Dispersal distances of crop plant pathogens display power-law relationships
(van der Plank 1960), and similar relationships have been suggested for plant
propagules (Harper 1977). Based on these observations, Kenkel and Irwin (1994)
hypothesized that the dispersal of diaspores and pathogens have fractal
properties. They suggested that Lévy or Cauchy flights (Mandelbrot 1982: § 32)
are appropriate models of dispersal. Species producing diaspores adapted for
long-distance dispersal (e.g. 'weeds') have a low fractal dimension. These
species advance through the landscape in large leaps, continually establishing
new colonies or epicenters (a 'guerilla' strategy). As a result, they display
highly patchy distributions at all spatial scales. Conversely, species lacking
adaptations for long-distance dispersal move through the landscape more
conservatively (a 'phalanx' strategy), with only occasional 'forays' to
establish new epicenters. These species have a higher fractal dimension,
resulting in less patchy, more continuous spatial distributions. If this model
is correct, outbreaks of pathogens having a low fractal dimension will be
difficult to predict, since new outbreaks will seem to appear from nowhere.

Shaw (1994) expands on these ideas, noting that classical dispersal
probability models are exponential (that is, all their moments are defined).
Exponential models assume that dispersal has a characteristic scale, implying
that long-distance dispersal is completely negligible. Exponential-based
simulation models result in a 'wave-expanding' dispersal pattern, where wave
velocity is proportional to the intrinsic population growth rate. However,
empirical studies typically demonstrate that gene flows are much greater than
those predicted by exponential models. More realistic models are obtained by
using dispersal probability distributions having infinite first and higher
moments. Shaw (1994, 1995) uses the Cauchy distribution (analogous to the
'Cauchy flight' of Mandelbrot 1982) to model dispersal. Cauchy-based models
produce patterns in which 'daughter foci' are continuously formed, so that
dispersal is best described as a disjoint set of locations (c.f. Kenkel and
Irwin 1993). Mayer and Atzeni (1993) used the Cauchy distribution to model
dispersal distance in the screwworm fly.

Wallinga (1995) modelled weed dynamics under the assumption that weed
populations are maintained at low densities (through tillage practices,
application of herbicides, and so forth). Under such a scenario, populations are
expected to display 'critical phenomena' (Grassberger 1983), resulting in their
dynamics and spatial pattern being scale-invariant. Fractal analysis
(correlation dimension) of a mapped point pattern of cleavers, a European weed,
confirmed the fractal (scale-invariant) nature of weed populations.

Collins and Glenn (1990) argue that competition and dispersal act together to
create fractal patterns in tall-grass prairie plant communities. They found
evidence of self-similarity in these grasslands (i.e. small-scale patterns are
repeated at larger spatial scales).

The hyperbolic distribution, because it lacks a characteristic scale,
describes the sizes of self-similar phenomena (Goodchild and Mark 1987). Meltzer
and Hastings (1992) examined the size distribution of grazed areas in Zimbabwe,
and related H to the relative stability of vegetation patches. Overall, they
found that increases in cattle density decreased patch stability. Using similar
methods, Hastings et al. (1982) found lower stability in earlier successional
patches. Kent and Wong (1982) used the size-frequency distribution of lakes to
estimate the fractal dimension of littoral zone habitat in the Precambrian
Shield of Ontario, while Hamilton et al. (1992) estimated terrain fractal
dimension based on lake size distributions in the Amazon and Orinoco river
floodplains. The hyperbolic distribution has also been fit to taxonomic systems
(Burlando 1990, 1993) and the size-distribution of seeds (Hegde et al. 1991).
Frontier (1987:359-367) discusses applications of fractal theory to
rank-frequency diagrams of the distribution of individuals among species.

Krummel et al. (1987) examined the fractal dimension of forest patches
('islands') using the perimeter-area method. They found that smaller forest
patches had lower mean D than larger patches. The transition zone from low to
high fractal dimension occurred at approximately 60-73 ha. They concluded that
small forest patches are the result of anthropogenic activities. This decrease
in landscape complexity with increasing anthropogenic activity was also reported
by O'Neill et al. (1988) and Turner and Ruscher (1988). De Cola (1989) used the
perimeter-area method to determine fractal dimensions of eight natural and
anthropogenic landscape-level classes in northern Vermont. Bian and Walsh (1993)
used two-dimensional semivariance to examine scale dependency in the
relationship between topography (elevation, slope angle, and slope aspect) and
reflectance/absorbance of vegetation at Glacier National Park, Montana. Studies
involving fractal dimension estimation of geomorphological features are
summarized in Goodchild and Mark (1987), Lam (1990) and Lam and Quattrochi
(1992).

A simplifying assumption of many classical ecological models is that habitats
are uniform, and that they vary linearly with distance. Some recent studies have
examined these assumptions and/or modified the classical models in light of the
recognized fractal nature of habitats. Scheuring (1991) modified the classical
species-area relationship model to include the fractal nature of vegetation.
Palmer (1992) modified the 'competition gradient' model of Czárán (1989) to
include fractal habitat complexity. He found that species coexistence increased
as landscape fractal dimension increased. Milne et al. (1992) examined mammalian
herbivore foraging in artificial fractal landscapes. They concluded that the
fractal nature of landscapes is an important determinant of resource utilization
rates. Milne (1992) examined the fractal geometry of landscapes from the
viewpoint of habitat fragmentation. He concluded that habitat fragmentation
affects ecosystem processes, and that this must be recognized in developing an
ecologically meaningful view of landscapes and habitats. Haslett (1994) found
that the fractal dimension of mountain meadow landscapes correlated well with
the abundance of syrphid flies, suggesting that more spatial heterogeneous
habitats may support more complex ecological communities. Additional potential
applications of fractal analysis to vegetation complexity are outlined by van
Hees (1994).

The spatial dependency of image elements (e.g. pixels) is referred to as
texture. A 'textural feature' is a combination of image elements that cannot be
individually differentiated (Musick and Grover 1990). A number of image
segmentation methods for the extraction of textural features are available
(Davis 1981; van Gool et al. 1985; Blacher et al. 1993). Fractal-based texture
methods overcome some of the problems inherent in classical resolution-sensitive
techniques (van Gool et al. 1985), and are particularly well-suited to complex
natural scenes (Keller et al. 1987; Pentland 1984).

Keller et al. (1989) describe a modified box-counting texture analysis
technique based on the probability density function. They characterized
simulated (Brodatz) textures in terms of fractal dimension and lacunarity. An
alternative box-counting method was proposed by Sarkar and Chaudhuri (1992). In
their method, each x,y coordinate has an associated third dimension
(z-coordinate) representing pixel intensity (e.g. gray shade). The box count is
determined as the number of cells in a column intercepted by the surface. Using
simulated textures, they found that their method was more computationally
efficient than that proposed by Keller et. al. (1989); see also Chaudhuri et al.
(1993) and Chaudhuri and Sarkar (1995). De Cola (1993) describes a hierarchical
grid method for the analysis of surface texture in remotely sensed images. It
was found that fractal dimension varied with scale, implying multifractal
behaviour.

Pentland (1984) developed a fractional Brownian motion (fBm) approach to
image texture analysis based on a modified Fourier algorithm. An analysis of
photographs of natural objects found that texture was more effective than
spectral properties in characterizing major image features. Keller et al. (1987)
used the fBm approach to examine interface complexity of vegetation/landform
types. Using the same approach, Dennis and Dessipris (1989) found that
anti-aliasing filtering techniques improved estimates of the fractal dimension
of 'natural' images, but had little effect on synthetic images.

Image analysis has also been used in medicine and cellular biology. For
example, Fortin et al. (1992) analyzed local and large-scale structures in
cardiac magnetic resonance images and bone x-rays. They provide detailed
descriptions of fBm image analysis models. Note that the methods for fractal
analysis of self-affine signals described by Schepers et al. (1992) can also be
used in image analysis.